Journal of Geometry

, 110:19 | Cite as

A characterization of the set of internal points of a conic in PG(2,q), q odd

  • Stefano Innamorati
  • Fulvio ZuanniEmail author


A point P not on a non-degenerate conic C in PG(2, q), q odd, is called internal to C if no tangent line to C contains P, external otherwise. The set of internal points of C is a \(\frac{q(q-1)}{2}\)-set of type \((0,\frac{q-1}{2},\frac{q+1}{2})\). In this paper, we classify all \(\frac{q(q-1)}{2}\)-sets of class [0, mn] having exactly two kinds of outer points.


Sets of class \([0, m, n]\) three character sets conics 

Mathematics Subject Classification

51E20 51E21 



We are grateful to the anonymous referee for useful suggestions which noticeably improved the presentation of the paper.


  1. 1.
    Coykendall, J., Dover, J.: Sets with few intersection numbers from singer subgroup orbits. Eur. J. Comb. 22, 455–464 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    De Clerck, F., De Feyter, N.: A characterization of the sets of internal and external points of a conic. Eur. J. Comb. 28, 1910–1921 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Durante, N.: On sets with few intersection numbers in finite projective and affine spaces. Electron. J. Comb. 21(4), 4–13 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Durante, N., Napolitano, V., Olanda, D.: On Quadrics of \(PG(3, q)\), Trends in Incidence and Galois Geometries: A Tribute to Giuseppe Tallini, Caserta, Dipartimento di Matematica. Seconda Università di Napoli 67–76, (2010).
  5. 5.
    Ferri, O.: I k-insiemi di classe \([0,\frac{q-1}{2},\frac{q+1}{2}]\) di un piano proiettivo di ordine dispari. Rend. Mat. 3, 33–41 (1983)Google Scholar
  6. 6.
    Hamilton, N., Penttila, T.: Sets of type (a, b) from subgroups of \(\Gamma \mathit{L}(1, \mathit{p}^\mathit{R})\). J. Algebr. Comb. 13, 67–76 (2001)Google Scholar
  7. 7.
    Hirschfeld, J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, New York (1998)zbMATHGoogle Scholar
  8. 8.
    Hirschfeld, J.W.P., Szonyi, T.: Sets in a finite plane with few intersection numbers and a distinguished point. Discrete Math. 97, 229–242 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Innamorati, S., Zuanni, F.: On sets of class \([1,q+1,2q+1]_2\) in \(PG(3,q)\). Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali. 96 (S2) (2018), Article number 6.
  10. 10.
    Innamorati, S., Zuanni, F.: On the parameters of two-intersection sets in \(PG(3,q)\). Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali. 96 (S2) (2018), Article number 7.
  11. 11.
    Penttila, T., Royle, G.F.: Sets of type \((m, n)\) in the affine and projective planes of order nine. Des. Codes Cryptogr. 6, 229–245 (1995)Google Scholar
  12. 12.
    Rodgers, M.: Cameron–Liebler line classes. Des. Codes Cryptogr. 68, 33–37 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tallini, G.: Graphic characterization of algebraic varieties in a Galois space. Teorie Combinatorie, Vol. II, Atti dei Convegni Lincei, 17, Roma, 3–15 settembre 1973 , pp. 153–165 (1976)Google Scholar
  14. 14.
    Tallini Scafati, M.: The \(k\) -set of type \((m,n)\) of an affine space \(A_{r,q}\). Rend. Mat. 1, 63–80 (1981)Google Scholar
  15. 15.
    Tallini Scafati, M.: The k-set of \(PG(r,q)\) from the character point of view. Finite Geom. Baker, C.A., Batten, L.M. (eds.) Marcel Dekker Inc., New York, pp. 321–326 (1985)Google Scholar
  16. 16.
    Thas, J.A.: A combinatorial problem. Geom. Dedic. 1, 236–240 (1973)CrossRefGoogle Scholar
  17. 17.
    Ueberberg, J.: On regular \(\{v, n\}\) -arcs in finite projective spaces. J. Comb. Des. 1, 395–409 (1993)Google Scholar
  18. 18.
    Zuanni, F.: On sets of type \((m, m+q)_2\) in \(PG(3, q)\). J. Geom. 108(3), 1137–1155 (2017). MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zuanni, F.: On two-intersection sets in \(PG(r,q)\). J. Geom. 109 (2) (2018), Article number 26.

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Authors and Affiliations

  1. 1.Department of Industrial and Information Engineering and EconomicsUniversity of L’AquilaL’AquilaItaly

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